I don't backstitch when piecing my quilts, and I don't use the small stitches idea either--I just chain stitch pieces as much as I can and clip the connecting threads in the middle, which leaves a little thread sticking out on each piece. I use my walking foot for piecing quilts and for most of my other sewing, because I then have less problem with the top fabric shifting forward a little.
I have a Bernina machine and walking foot, and no one ever mentioned that I shouldn't backstitch, so I do it when making garments.
I apply bindings by machine by stitching them to the front of the quilts (using that walking foot) and after turning them to the back, stitching in the ditch from the front. They do not always look even on the back, but I don't worry about it. I cut 2-1/2 inch strips for bindings, and use a generous 1/4 inch seam to apply to the front of the quilts.